To factor the expression [tex]\( x^8 - 81 \)[/tex] completely, let's follow the steps to break it down.
1. Recognize the structure of the polynomial:
The expression [tex]\( x^8 - 81 \)[/tex] can be viewed as a difference of squares:
[tex]\[
x^8 - 81 = (x^4)^2 - 9^2
\][/tex]
Using the difference of squares formula [tex]\( a^2 - b^2 = (a - b)(a + b) \)[/tex], we get:
[tex]\[
x^8 - 81 = (x^4 - 9)(x^4 + 9)
\][/tex]
2. Factor the term [tex]\( x^4 - 9 \)[/tex]:
Notice again that [tex]\( x^4 - 9 \)[/tex] can be written as a difference of squares:
[tex]\[
x^4 - 9 = (x^2)^2 - 3^2
\][/tex]
Applying the difference of squares formula again, we have:
[tex]\[
x^4 - 9 = (x^2 - 3)(x^2 + 3)
\][/tex]
3. Rewrite the expression:
Substitute the factored form of [tex]\( x^4 - 9 \)[/tex] back into the original expression:
[tex]\[
x^8 - 81 = (x^2 - 3)(x^2 + 3)(x^4 + 9)
\][/tex]
So, the completely factored form of [tex]\( x^8 - 81 \)[/tex] is:
[tex]\[
(x^2 - 3)(x^2 + 3)(x^4 + 9)
\][/tex]
Therefore, the correct choice among the options provided is:
[tex]\[
\left(x^2 - 3 \right)\left(x^2 + 3 \right)\left(x^4 + 9 \right)
\][/tex]
